Does the PoGo app have to be open but not necessarily the active one on the screen to work?
pottypotamus No. The whole point of Adventure Sync is that you don’t have to have the app open so you can save battery life but still hatch eggs 
OhLookHeKickedTheBall That may br a glitch for you, but it is not the case for everyone. I have no issues with Adventure Sync syncing correctly whether the app is open, closed or running in the background.
If you are having Adventure Sync issues, make sure you have taken all of the following steps: support.pokemongo.nianticlabs.com/hc/en-us/articles/360002025707-Adventure-Sync